Boolean logic and gates · CA Midterm Trainer

Basic gates

AND requires all inputs high; OR requires at least one; NOT inverts. XOR is high for unequal inputs and XNOR for equal inputs.

Symbols and expressions of the basic gates.

NAND=NOT(AND) and NOR=NOT(OR). Each is functionally complete: one gate type can build NOT, AND, and OR, and therefore any Boolean function.

NOT, AND, and OR from NAND only.

NAND from four NORs (Midterm 2024).

NOT(A AND B)=(NOT A) OR (NOT B); NOT(A OR B)=(NOT A) AND (NOT B). Moving inversion through a gate swaps AND and OR.

Pushing a bubble through a gate swaps AND↔OR.

For two inputs list 00, 01, 10, and 11. Comparing every row is a reliable way to prove two expressions or circuits equivalent.

Build it yourself: OR from NAND

An ideal output depends only on current inputs. Feedback and memory make a circuit sequential.

A gate does not update instantly. Worst-case delay is the sum of delays along the longest causal path—the critical path.

A path delay is the sum of gate delays; the critical path is the longest.

SELF-CHECK

When is XOR=1?

When inputs differ.

Is XNOR universal?

No, not by itself.

Name the universal gates.

NAND and NOR.

What does the critical path determine?

Worst-case propagation delay.